Degree-truncated choosability of planar graphs
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arXiv
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| Auteurs principaux: | , , , |
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| Format: | Preprint |
| Publié: |
2024
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| _version_ | 1866908448405323776 |
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| author | Jiang, Yiting Xu, Huijuan Xu, Xinbo Zhu, Xuding |
| author_facet | Jiang, Yiting Xu, Huijuan Xu, Xinbo Zhu, Xuding |
| contents | Assume $G$ is a graph and $k$ is a positive integer. Let $f:V(G)\to \mathbb{N}$ be defined as $f(v)=\min\{k,d_G(v)\}$. If $G$ is $f$-choosable, then we say $G$ is degree-truncated $k$-choosable. Answering a question of Richter, it was proved in [Zhou,Zhu,Zhu, Degree-truncated choice number of graphs, arXiv:2308.15853] that there exists a 3-connected non-complete planar graph that is not degree-truncated 7-choosable, and every 3-connected non-complete planar graph is degree-truncated 16-choosable. This paper improves the bounds, and proves that there exists a 3-connected non-complete planar graph that is not degree-truncated 8-choosable, and that every 3-connected non-complete planar graph is degree-truncated $12$-choosable. |
| format | Preprint |
| id |
arxiv_https___arxiv_org_abs_2406_06035 |
| institution | arXiv |
| publishDate | 2024 |
| record_format | arxiv |
| spellingShingle | Degree-truncated choosability of planar graphs Jiang, Yiting Xu, Huijuan Xu, Xinbo Zhu, Xuding Combinatorics Assume $G$ is a graph and $k$ is a positive integer. Let $f:V(G)\to \mathbb{N}$ be defined as $f(v)=\min\{k,d_G(v)\}$. If $G$ is $f$-choosable, then we say $G$ is degree-truncated $k$-choosable. Answering a question of Richter, it was proved in [Zhou,Zhu,Zhu, Degree-truncated choice number of graphs, arXiv:2308.15853] that there exists a 3-connected non-complete planar graph that is not degree-truncated 7-choosable, and every 3-connected non-complete planar graph is degree-truncated 16-choosable. This paper improves the bounds, and proves that there exists a 3-connected non-complete planar graph that is not degree-truncated 8-choosable, and that every 3-connected non-complete planar graph is degree-truncated $12$-choosable. |
| title | Degree-truncated choosability of planar graphs |
| topic | Combinatorics |
| url | https://arxiv.org/abs/2406.06035 |